In this paper, following the work of Huneke and Wiegand [1]}, we study the (non) vanishing of local cohomology modules, and investigate depth and torsion properties of tensor products of modules. The new advantage we have is that we work with local cohomology functors with respect to specialization-closed subsets of \(\operatorname{Spec}R\) ; this allows us to generalize the results of Huneke and Wiegand in this direction, as well as various results from the literature, especially those stated in terms of Serre's conditions.
